Hilbert Modular Forms

Mod P and P-adic Aspects

Buch | Softcover

2005
American Mathematical Society (Verlag)
978-0-8218-3609-5 (ISBN)

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Studies Hilbert modular forms in characteristic $p$ and over $p$-adic rings. This title describes the kernel and image of the $q$-expansion map and prove the existence of filtration for Hilbert modular forms; define operators $U$, $V$ and $/Theta_/chi$; and study the variation of the filtration under these operators.

We study Hilbert modular forms in characteristic $p$ and over $p$-adic rings. In the characteristic $p$ theory we describe the kernel and image of the $q$-expansion map and prove the existence of filtration for Hilbert modular forms; we define operators $U$, $V$ and $/Theta_/chi$ and study the variation of the filtration under these operators. Our methods are geometric - comparing holomorphic Hilbert modular forms with rational functions on a moduli scheme with level-$p$ structure, whose poles are supported on the non-ordinary locus.In the $p$-adic theory we study congruences between Hilbert modular forms. This applies to the study of congruences between special values of zeta functions of totally real fields. It also allows us to define $p$-adic Hilbert modular forms 'a la Serre' as $p$-adic uniform limit of classical modular forms, and compare them with $p$-adic modular forms 'a la Katz' that are regular functions on a certain formal moduli scheme. We show that the two notions agree for cusp forms and for a suitable class of weights containing all the classical ones. We extend the operators $V$ and $/Theta_/chi$ to the $p$-adic setting.

Introduction Notations Moduli spaces of abelian varieties with real multiplication Properties of $/mathcal{G}$ Hilbert modular forms The $q$-expansion map The partial Hasse invariants Reduceness of the partial Hasse invariants A compactification of $/mathfrak{M} (k,/mu_{pN})^{/rm{Kum}}$ Congruences mod $p^n$ and Serre's $p$-adic modular forms Katz's $p$-adic Hilbert modular forms The operators $/Theta_{/mathfrak{P},i}$ The operator $V$ The operator $U$ Applications to filtrations of modular forms Theta cycles and parallel filtration (inert case) Functorialities Integrality and congruences for values of zeta functions Numerical examples Comments regarding values of zeta functions References.

Erscheint lt. Verlag	30.1.2005
Reihe/Serie	Memoirs of the American Mathematical Society
Zusatzinfo	illustrations
Verlagsort	Providence
Sprache	englisch
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	0-8218-3609-9 / 0821836099
ISBN-13	978-0-8218-3609-5 / 9780821836095
Zustand	Neuware